Construction of Generically Ordinary Families of Hyperelliptic Curves

Abstract: Katz conjectured in a 2018 lecture that the family of curves $y<sup>2=x<sup>d-dx+t$ over the $t$-line is generically ordinary for all sufficiently large primes $p$. We prove that, for every $g\ge 2$ and every nonzero algebraic integer $α$, the genus-$g$ families $C_α: y<sup>2=x<sup>d+αx+t$ where $d\in{2g+1, 2g+2}$ are generically ordinary at every prime $p&gt;P<sup>+(d)$, provided that $α$ is nonzero modulo every prime above $p$. The bound $P<sup>+(d)=d<sup>2-4d+2$ if $d$ is odd, and $P<sup>+(d)=(d<sup>2-3d+2)/2$ if $d$ is even.

• The paper proves that hyperelliptic curves y²=xᵈ+αx+t are generically ordinary for primes exceeding the explicit bound P⁺(d), confirming Katz's conjecture.
• It introduces a constructive method using the Hasse–Witt matrix determinant and refined p-adic analysis to detect ordinarity.
• The results extend known findings to arbitrary genus, providing new tools for studying monodromy and moduli spaces over finite fields.

Construction of Generically Ordinary Families of Hyperelliptic Curves

Introduction and Motivation

The study addresses a conjecture formulated by Katz regarding the generic ordinarity of certain families of hyperelliptic curves over finite fields, specifically those of the form $y^2 = x^d - d x + t$, as the characteristic $p$ increases. Ordinarity, detected via the determinant of the Hasse–Witt or Cartier–Manin matrix and reflected in the Newton polygon of the Jacobian, is a critical property with implications for the arithmetic and moduli of algebraic curves. The result responds to open questions about the prevalence of ordinary fibers in parameterized families of curves with large monodromy, particularly those with explicit equations.

Main Results and Theoretical Contributions

The principal theorem proves that for any $g \ge 2$ and nonzero algebraic integer $\alpha$, the families of genus-$g$ hyperelliptic curves $C_\alpha: y^2 = x^d + \alpha x + t$ with $d \in \{2g+1, 2g+2\}$ are generically ordinary at all primes $p > P^+(d)$, provided $\alpha$ remains nonzero modulo primes above $p$. The explicit bound $p$0 is given by $p$1 when $p$2 is odd and by $p$3 when $p$4 is even. This bound is shown to be sharp in light of counterexamples at $p$5.

The strong numerical guarantee—explicit, uniform in $p$6, and covering an infinite family of curves—addresses a gap left open by previous partial results and by Katz's conjecture. The proof is constructive and predicated on a new approach to analyzing the monodromy and the reduction mod $p$7 of Hasse–Witt matrices in families where the monodromy group is large but the determinant does not reduce to a trivial permutation structure, as in the cases previously considered by Miller.

In particular, the result generalizes known phenomena for elliptic curves ($p$8) and genus-2 cases, previously understood via ad hoc or computational arguments, to arbitrary genus. The methods also extend to two-parameter families, confirming generic ordinarity in broader moduli and illustrating the universality of the constructed bounds.

Technical Approach

A central technical innovation is the evaluation and factorization of a determinant $p$9 associated with the leading coefficient of the Hasse–Witt matrix, ultimately relating vanishing criteria for ordinarity to explicit number-theoretic information about the monodromy and parameterization. The determinant is shown to admit a product formula with all possible prime divisors bounded by $g \ge 2$0, and its non-vanishing for $g \ge 2$1 is established through a careful combinatorial and $g \ge 2$2-adic analysis. These results exploit refined manipulations of binomial and multinomial coefficients, connections to Morse polynomials, and work in the context of parameterized families over the ring of integers of the number field generated by $g \ge 2$3.

The strategy involves reducing the question of generic ordinarity in the family to the non-vanishing mod $g \ge 2$4 of the determinant of the coefficient matrix $g \ge 2$5, whose entries encode the coefficients of $g \ge 2$6 in $g \ge 2$7. The determinant's leading coefficient is shown, via congruences and product expansions, to avoid zero modulo all sufficiently large $g \ge 2$8.

The paper further clarifies the relationship between the family $g \ge 2$9 and the parameter $\alpha$0, establishing isomorphisms over finite extensions and providing an explicit dependence of the ordinary locus on parameterizations in the moduli space.

Implications and Outlook

The result provides a definitive answer to Katz's conjecture for the broad class of hyperelliptic curves $\alpha$1 and confirms the expected ubiquity of ordinary fibers in families with large monodromy when the characteristic $\alpha$2 is large. Practically, this facilitates the construction of curves with predictable $\alpha$3-torsion behavior and informs cryptographic applications as well as the geometry of moduli spaces over finite fields.

Theoretically, the proof techniques offer new tools for analyzing monodromy and Newton polygon stratifications in higher-dimensional families, prompting extensions to more general classes of curves, higher-rank monodromy, and other stratifications. The explicit product bounds derived for determinantal divisors may aid in sieving methods for explicit curve constructions and in the study of unlikely intersections within $\alpha$4-rank stratifications.

Beyond the immediate context, this approach solidifies the bridge between explicit arithmetic geometry and the asymptotic behavior of moduli spaces, suggesting routes to analyze genericity phenomena in other settings such as non-hyperelliptic curves, higher-dimensional varieties, and families with more intricate ramification structures.

Conclusion

The paper establishes explicit, sharp bounds on the characteristic $\alpha$5 above which the families $\alpha$6 are generically ordinary, confirming Katz's conjecture for all genera $\alpha$7 and a wide class of parameterizations. This advances the classification of ordinary loci in moduli, enriches the theoretical toolbox for studying monodromy and $\alpha$8-rank, and enables new explicit constructions relevant to arithmetic geometry and its applications. The methods and results are poised to influence further advances on both practical and foundational questions concerning the distribution of the ordinary locus in families of curves over finite fields.

Reference: "Construction of Generically Ordinary Families of Hyperelliptic Curves" (2606.31783)